A book-embedding of a graph G is an embedding of vertices of G along the spine of a book, and edges of G on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the page number. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectured that its page number can be made arbitrarily large [JCT, 1979, 320-331]. Heath [FOCS84] found that planar graphs admit a seven-page book embedding.Later, Yannakakis proved that four pages are necessary and sufficient for planar graphs in [STOC86]. Recently, Bekos et al. [STACS14] described an O(n 2 ) time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an O(n 2 ) time algorithm.

The {K 2 , C n }-factor of a graph is a spanning subgraph whose each component is either K 2 or C n . In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the {K 2 , C 2i+1 |i ≥ 2}-factor for any graph is obtained, which answers a problem due to Gao and Wang (J. Oper. Res. Soc. China (2021),

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